What Are the Frequency Components of an FM Wave?

[H_man]

Homework Statement

I wanted to show that a particular wave is composed of the 4 frequencies

W = W₀ +/- W₁ +/- W₂

The equation for this FM wave is y(t) = Acos(W₀t+W₁COS(W₂t)t )

I tried showing the different frequency components by performing a Fourier transform on the equation y(t). However I got a bit stuck with integrating the cosine term as part of the exponential function. I thought about expanding it as a Taylor series but that doesn't really help.

If this was a practical problem I would just use Matlab and a DFT and hey presto. But trying to do this analytically is posing a problem.

Expanding in terms of trigonometric identities to show the point also doesn't seem to get me closer to a solution.

y(t) = Acos(W₀t+W₁COS(W₂t)t ) =

= A{cos(W₀t)cos(W₁COS(W₂t)t ) -
sin(W₀t)sin(W₁sin(W₂t)t )}

we are given that W₁t<<1
and W₁²t -> 0
so if we assume cos(W₁COS(W₂t)t )~1

Then we have:

= A{cos(W₀t)-sin(W₀t)W₁cos(W₂t)t )}

= A{cos(W₀t) - sin(W₀t+W₂t)W₁ +
W₁ cos(W₀t)sin(W₂t)

But I don't see a solution from here...

 

[H_man]

Its okay. I just found the solution. It involves Bessel Functions...